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Computer Graphics
Viewing
What Are Projections?
Our 3-D scenes are all specified in 3-D world
coordinates
To display these we need to generate a 2-D
image - project objects onto a picture plane
So how do we figure out these projections?
Picture Plane
Objects in
World Space
Converting From 3-D To 2-D
Projection is just one part of the process of
converting from 3-D world coordinates to a 2-
D image
Clip against
view volume
Project onto
projection
plane
Transform to
2-D device
coordinates
3-D world
coordinate
output
primitives
2-D device
coordinates
Types Of Projections
There are two broad classes of projection:
Parallel: Typically used for architectural and
engineering drawings
Perspective: Realistic looking and used in
computer graphics
Perspective Projection Parallel Projection
Types Of Projections (cont…)
For anyone who did engineering or technical
drawing
Parallel Projections
Some examples of parallel projections
Orthographic Projection
Isometric Projection
Isometric Projections
Isometric projections have been used in
computer games from the very early days of
the industry up to today
Q*Bert Sim City Virtual Magic Kingdom
Perspective Projections
Perspective projections are much more
realistic than parallel projections
Perspective Projections
There are a number of different kinds of
perspective views
The most common are one-point and two
point perspectives
One Point Perspective
Projection
Two-Point
Perspective
Projection
Elements Of A Perspective Projection
Virtual
Camera
The Up And Look Vectors
The look vector indicates the direction in which the
camera is pointing
The up vector determines how the camera is rotated
For example, is the camera held vertically or
horizontally
Up vector Look vector
Position
Projection of
up vector
Contents
In today’s lecture we are going to have a look
at:
Transformations in 3-D
How do transformations in 3-D work?
3-D homogeneous coordinates and matrix based
transformations
Projections
History
Geometrical Constructions
Types of Projection
Projection in Computer Graphics
3-D Coordinate Spaces
Remember what we mean by a 3-D
coordinate space
x axis
y axis
z axis
P
y
z
x
Right-Hand
Reference System
Translations In 3-D
To translate a point in three dimensions by
dx, dy and dz simply calculate the new points
as follows:
x’ = x + dx y’ = y + dy z’ = z + dz
(x’, y’, z’)
(x, y, z)
Translated Position
Scaling In 3-D
To sale a point in three dimensions by sx, sy
and sz simply calculate the new points as
follows:
x’ = sx*x y’ = sy*y z’ = sz*z
(x, y, z)
Scaled Position
(x’, y’, z’)
Rotations In 3-D
When we performed rotations in two
dimensions we only had the choice of rotating
about the z axis
In the case of three dimensions we have more
options
Rotate about x – pitch
Rotate about y – yaw
Rotate about z - roll
Rotations In 3-D (cont…)
x’ = x·cosθ - y·sinθ
y’ = x·sinθ + y·cosθ
z’ = z
x’ = x
y’ = y·cosθ - z·sinθ
z’ = y·sinθ + z·cosθ
x’ = z·sinθ + x·cosθ
y’ = y
z’ = z·cosθ - x·sinθ
The equations for the three kinds of rotations
in 3-D are as follows:
Homogeneous Coordinates In 3-D
Similar to the 2-D situation we can use homogeneous
coordinates for 3-D transformations - 4 coordinate
column vector
All transformations can
then be represented
as matrices
1
z
y
x
x axis
y axis
z axis
P
y
z
x
P(x, y, z) =
3D Transformation Matrices
1000
100
010
001
dz
dy
dx
1000
000
000
000
z
y
x
s
s
s
1000
0cos0sin
0010
0sin0cos
Translation by
dx, dy, dz Scaling by
sx, sy, sz
1000
0cossin0
0sincos0
0001
Rotate About X-Axis
1000
0100
00cossin
00sincos
Rotate About Y-Axis Rotate About Z-Axis
Remember The Big Idea
Summary
In today’s lecture we looked at:
Transformations in 3-D
Very similar to those in 2-D
Projections
3-D scenes must be projected onto a 2-D image
plane
Lots of ways to do this
Parallel projections
Perspective projections
The virtual camera
Who’s Choosing Graphics?
A couple of quick questions for you:
Who is choosing graphics as an option?
Are there any problems with option time-
tabling?
What do you think of the course so far?
Is it too fast/slow?
Is it too easy/hard?
Is there anything in particular you want to cover?
3D Transformations
• Same idea as 2D transformations
– Homogeneous coordinates: (x,y,z,w)
– 4x4 transformation matrices
w
z
y
x
ponm
lkji
hgfe
dcba
w
z
y
x
'
'
'
'
wz
yx
wz
yx
1000010000100001
'
''
w
z
y
x
t
t
t
w
z
y
x
z
y
x
1000
100
010
001
'
'
'
Identity
Translation
Translation
W=1
w
z
y
x
s
s
s
w
z
y
x
z
y
x
1000
000
000
000
'
'
'
Scale
Scaling
wz
yx
wz
yx
1000010000100001
'
''
Identity
Column Vector Representation
Row Vector Representation
x’ y’ z’ 1 = x , y, z ,1
Rotation
wz
yx
wz
yx
1000010000cossin00sincos
'
''
Rotate around Z axis:
Column Vector Representation
Row Vector Representation
Rotation
Rotate around Y axis:
w
z
y
x
w
z
y
x
1000
0cos0sin
0010
0sin0cos
'
'
'
Column Vector Representation
Row Vector Representation
Rotation
Rotate around X axis:
wz
yx
wz
yx
10000cossin00sincos00001
'
''
Column Vector Representation
Row Vector Representation
Projection
Representing a three-dimensional object or scene in 2-
dimensional objects onto the 2-dimensional view plane.
There are 2 types of projections.
Parallel Projection
Perspective Projection
Categories
of
Projection
Parallel Projection
Orthographic Projection
Axonometric Projection
Isometric Projection
Dimetric Projection
Trimetric Projection
Oblique Projection
Cavalier Projection
Cabinet Projection
Perspective projection
Single-Point Projection
Two-Point Projection
Three-Point Projection
Parallel Projection
Parallel Projection preserves relative proportions of objects but does not
produce the realistic views
Perspective Projection
Perspective Projection produce the realistic views but does not
preserves relative proportions of objects
View Plane
Front View
Orthographic Parallel Projection
Axonometric Ortho…
Top,
Side,
rear
(far) :
Elevati
ons
&
Top
is
called
Plan
View
Isometric Projection of an object
onto a viewing plane
Oblique Parallel Projection
Cavalier Projections of the unit cube
Cabinet Projections of the Unit Cube
Types of Perspective Projections
Logical Relationship among the various
types of projections
Transformation Matrix for general Parallel
Projection (on XY plane)
(x1,y1,
Matrix Representation
Transformation Matrix for general Oblique
Projection (on XY plane)
Transformation Matrix for Perspective
Projection (on XY plane)
3D Clipping
HiDdEn SuRfAcE rEmOvAl
Example
Original Pyramid After Hidden Surface
Removal
cAtEgOrIeS oF hIDdEn SuRfAcE rEmOvAl
• oBJECT sPACE mETHOD
• iMAGE sPACE mETHOD
Hidden Surface Removal/ Visible Surface Detection
Object Space Method
Robert’s Algorith
m
Back Face
Removal/Detecti
on Method
Painter’s Algorith
m (Depth Sort)
Image Space Method
Area Subdivisi
on Method (Warnoc
k’s)
Octree method
Depth Buffer
Method (Z-Buffer
Scan Line
Algortihm
Ray Tracing Method
oThErS
• fLOATING hORIZON aLGORITHM
• bINARY sPACE pARTITIONING
bAcK fAcE rEmOvAl mEtHoD • Back face means the surface of the polygon which in
not visible in projection. So we have to remove this surface from projection.
• It is used for identifying back faces of a polyhedron is base on the inside-outside test.
• Back face removal algorithm will be applied on plane polygons.
A point (x,y,z) is inside a polygon surface with plane parameters A, B, C and D if
Ax + By + Cz + D < 0
• Figure : Vector V in the viewing direction and a back-face normal vector N of a polyhedron
N= (A,B,C)
Camera
V
• The normal vector N to a polygon surface, which has Cartesian Components (A,B,C).
• If V is a vector in the viewing direction from the eye or camera position, then this position is a back face if
V.N > 0
• If the dot product is positive, we can say that the polygon faces towards the viewer, otherwise it faces away and should be removed.
• In case, if object description has been converted to projection coordinates and our viewing direction is parallel to the viewing Zv
axis, then V = (0,0,Zv) and V.N = Zv C
• To consider the sign of C , the z component of the normal vector N. Now if the Z component is positive, then the polygon faces towards the viewer, if negative it faces away .
Yv
Xv
Zv
N= (A,B,C)
V
Example I
EYE
EYE
FRONT
BACK
Example II Y
X
Z
C D
B A
A B
D C
A B
D C
A B
D C
A B
D C
Back-face Detection
Method Direction for
Faces
ABC BDC
ACD ADB
DePtH cOmPaRiSiOn
Z
Y
X
B(x2,y2,z2)
A(x1,y1,z1)
P Parallel Projection
If A and B are not on the same projection line then no point hide the other point.
If A and B are on the same projection line then in case of parallel projection on xy plane if x1=x2 and y1 = y2 then A and B are on same plane. If Z1<Z2 then A point hide B.
Frame Buffer
Z
Y
X
B(x2,y2,z2)
A(x1,y1,z1)
P Perspective Projection
Z-Buffer (Depth Buffer) Algorithm
Z-Buffer (Depth Buffer) Algorithm
Y
X
Z
(X , Y ) Result At view plane position (x,y),surface S1 has the smallest depth from the view plane and so is visible at that position
Z-Buffer Algorithm
• Initialize frame buffer to background colour.
• Initialize z-buffer to minimum z value.
• Scan convert each polygon in arbitrary order.
• For each(x,y) pixel, calculate depth ‘z’ at that pixel(z(x,y)).
• Compare calculated new depth z(x,y) with value previously stored in z-buffer at that location z(x,y).
• If z(x,y)>z(x,y), then write the new depth value to z-buffer and update frame buffer.
• Otherwise, no action is taken.
• The plane polygon define a surface or plane whose equation can be written as
Ax + By + Cz + D = 0 Depth value for a surface position (x.y) are calculated from the plane equation for each surface z = -Ax –By-D C
Z’ = -A (x+1) – By –D
C
From position (x,y) on a scan line, the next position across the line has coordinates (x+1,y) and the position immediately below on the next line has coordinates (x,y-1)
Top Scan Line
Y Scan Line
Bottom Scan Line
Fig: Scan Lines intersecting a polygon surface
Top Scan Line
Y Scan Line
Bottom Scan Line
Fig: Intersecting positions on successive scan line along a Left Polygon
edge
Y-1 Scan Line
X X’
Scan Line Method
Scan Line 1
Scan Line 2
S1 S2
a
b
c
d g
h
f
e
Scan Lines crossing the projection of two surface, S1 and S2 in the view plane. Dashed lines indicates the boundaries of hidden surfaces.
Scan Line1 S1: edge ab & bc , s2: eh &ef Scan Line2 S1: ad & eh , s2 :bc &fg S1 & s2 : eh & bc Depth value S1<S2 then s1 is visible
Intersecting and cyclically overlapping surfaces
that alternately obscure(Unclear) one anther
Intersecting and cyclically overlapping surfaces
that alternately obscure(Unclear) one anther
Painter Algorithm (Depth Method)
• Using both image space and object space operations, the depth-sorting method performs the following basic functions:
– Surfaces are sorted in order of decreasing depth.
– Surfaces are scan converted in order, starting with the surface of greatest depth.
Used : Oil Painting ,an artist first paints the
background color
tEST -I
A
B
Surface A and B with no overlapping in Z direction
ZAmax
ZAmin
ZBmax
ZBmin Xv
Zv Yv
First of all find the overlapping in z axis from the sorted list of polygons according to z value
tEST -II
A
B
Surface A and B with no overlapping in Y direction
YAmax
YAmin
YBmax
YBmin
Xv
Zv Yv
First of all find the overlapping in y axis from the sorted list of polygons according to y value
tEST -III
B
Surface A and B that do not overlap in x direction
XBmax XBmin XAmax XAmin
A
Xv
Zv Yv
First of all find the overlapping in x axis from the sorted list of polygons according to x value
tEST -IV
Surface A completely overlaps the background plane B
Zv
Yv
Xv
A
B
If some plane overlapping with background plane then we have test whether the plane completely hide the background plane or not.
tEST -V
Surface A does not hide B
Zv
Yv
Xv
A
B
If some plane overlapping with background plane then we have test whether the plane completely in front of background plane but not hiding the background plane.
Others
Three surfaces entered into the sorted surface list in the order A,B,C should be reordered A, C, B.
Zv
Yv
Xv
A C B
Binary Space Portitioning Tree (BSP) Method
D
C
A
B Back
Front Back
Front
P1
P2
View P1
FRONT P2 A , B
BACK P2 C, D
A region of space is
partitioned with two planes
P1 and P2.
P1
Back
Front
Front Front Back
Back
P2 P2
A C B D
BSP Tree Representation
aReA-sUbDiViSiOn MeThOd (WoRnOcK’s AlGoRiThM)
S
Initial Viewing Area Subdivision of Viewing Area
• The relationship between projection each polygon and the area of interest is checked for four possible relationships :
Surrounding Surface
Overlapping OR Intersecting Surface
Inside OR Contained Surface
Outside OR Disjoint Surface
Case I – Surrounding Surface P
S
Polygon that completely surrounds the screen area is called surrounding surface .
If polygon is surrounding the screen area , color the all pixels of screen area as color of screen.
S
Case II – Outside or Disjoint Surface
P S
Polygon that is completely outside the screen area.
If all polygon comes under the category of outside polygon, color all pixel of viewing screen as background color.
S
Case III – Contained or Inside Surface
P
S
Polygon that is completely inside the screen area.
If polygon in inside the screen area, we scan convert that area and the remaining area of screen will colored with background color.
S
Case IV – Intersecting or Overlapping Surface
P
S
Polygon that intersect the
screen area S.
Others
AREA
Inside Polygon
Intersecting
Polygon
Surrounding
Polygon
X
Y
Others
AREA
Intersecting
Polygon
Surrounding
Polygon
X
Y
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